<p>Statistical modeling aims at constructing a mathematical model for an existing data set. As a comprehensive concept, statistical modeling leads to a wide range of interesting problems. Modern parametric models, such as deepnets, have achieved remarkable success in quite a few application areas with massive data. Although being powerful in practice, many fitted over-parameterized models potentially suffer from losing good statistical properties. For this reason, a new framework named the Auto-modeling (AM) framework is proposed. Philosophically, the mindset is to fit models to future observations rather than the observed sample. Technically, choosing an imputation model for generating future observations, we fit models to future observations via optimizing an approximation to the desired expected loss function based on its sample counterpart and what we call an adaptive {\it duality function}.</p>
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<p>The first part of the dissertation (Chapter 2 to 7) focuses on the new philosophical perspective of the method, as well as the details of the main framework. Technical details, including essential theoretical properties of the method are also investigated. We also demonstrate the superior performance of the proposed method via three applications: Many-normal-means problem, $n < p$ linear regression and image classification.</p>
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<p>The second part of the dissertation (Chapter 8) focuses on the application of the AM framework to the construction of linear regression models. Our primary objective is to shed light on the stability issue associated with the commonly used data-driven model selection methods such as cross-validation (CV). Furthermore, we highlight the philosophical distinctions between CV and AM. Theoretical properties and numerical examples presented in the study demonstrate the potential and promise of AM-based linear model selection. Additionally, we have devised a conformal prediction method specifically tailored for quantifying the uncertainty of AM predictions in the context of linear regression.</p>
| Identifer | oai:union.ndltd.org:purdue.edu/oai:figshare.com:article/23729751 |
| Date | 25 July 2023 |
| Creators | Yiran Jiang (16632711) |
| Source Sets | Purdue University |
| Detected Language | English |
| Type | Text, Thesis |
| Rights | CC BY 4.0 |
| Relation | https://figshare.com/articles/thesis/Toward_a_Theory_of_Auto-modeling/23729751 |
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